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Chapter 4: Problem 13

A sphere of radius \(r\) and mass \(m\) has a linear velocity \(v_{0} \mathrm{~m} / \mathrm{s}\) directed to the left and no angular velocity as it is placed on a horizontal platform moving to the right with a constant velocity \(10 \mathrm{~m} / \mathrm{s}\). If after sliding on the platform the sphere is to have no linear velocity relative to the ground as it starts rolling on the platform without sliding. The coefficient of kinetic friction between the sphere and the platform is \(\mu_{k}\) Determine the required value of \(v_{0}\) in \(\mathrm{m} / \mathrm{s}\).

Short Answer

Expert verified
The required value of \(v_{0}\) is 0 m/s.

Step by step solution

01

Understanding Initial Conditions

The problem describes a sphere with a linear velocity \(v_0\), moving left, and initially without angular velocity. Simultaneously, the platform moves rightward with a velocity of 10 m/s.
02

Final Desired Conditions

When the sphere starts rolling without slipping on the platform, it should have zero linear velocity relative to the ground. This implies the sphere's center of mass velocity equals the platform's velocity, 10 m/s, to the right.
03

Condition for Rolling Without Slipping

For rolling without slipping, the relationship \(v = r\omega\) must hold, where \(v\) is the linear velocity of the sphere relative to the platform, \(r\) is the radius, and \(\omega\) is the angular velocity of the sphere.
04

Apply Relative Velocity Concept

The final velocity of the sphere relative to the platform (\(v_{final}\)) is 0. Thus, \(v + 10 = 0\) implies \(v = -10\ m/s\) relative to the initial condition where the sphere moves left.
05

Friction's Role in Changing Sphere's Motion

Kinetic friction \(f_k = \mu_k mg\) acts opposite to the direction of relative slip, providing both translational and rotational acceleration. It stops the linear slip of the sphere and starts the rotation.
06

Kinematics Equation for Translational Acceleration

The net force due to friction induces an acceleration \(a = -\mu_k g\) on the sphere. Using \(v^2 = u^2 + 2as\) for final relative velocity \(v_{final} = -10\ m/s\) and initial \(v_{0} = -v + 10 = v - 10\), solve for \(v_{0}\).
07

Kinematics for Angular Motion

The frictional force also causes angular acceleration \(\alpha = \frac{f_k}{I} r = \frac{5\mu_k g}{2r}\), where \(I = \frac{2}{5}mr^2\) is moment of inertia. For rolling without slipping, \(\omega = \frac{v}{r}\), relate these to find the effect of friction over time.
08

Solution Calculation

Substitute known values and solve required velocities. From the velocity equations and the conditions, \(v_{0} = 0\) as rolling without slipping results without an initial opposite velocity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinetic Friction
Kinetic friction plays a crucial role in this exercise.It is the force that opposes the relative motion of two surfaces in contact.In this case, it is the force that acts to stop the sliding motion of the sphere on the platform.Once the sphere is placed on the platform with an initial linear velocity and no angular velocity, kinetic friction becomes indispensable in altering the sphere's motion.
  • The force of kinetic friction (f_k = \mu_k mg) is applied in the opposite direction to the sphere's slip.
  • This force reduces the sphere's linear velocity while simultaneously inducing rotational motion.
  • Kinetic friction is proportional to the normal force and depends on the coefficient of kinetic friction \(\mu_k\u0011\).
Thus, kinetic friction is a transforming force, effectively converting the linear slip into rotational motion, allowing the sphere to progress into rolling without slipping.
Angular Velocity
Angular velocity is a measure of how quickly the object is rotating around its axis. It is given in radians per second and is a fundamental part of understanding how the sphere begins to roll without slipping.Initially, the sphere has no angular velocity, but as kinetic friction acts upon it, the sphere begins to spin or rotate.
  • The sphere's angular velocity \(\omega\) is closely related to the frictional force that generates rotational acceleration over time.
  • Using the formula \(\alpha = \frac{5\mu_k g}{2r}\), we see angular acceleration \(\alpha\) arises due to the friction.
  • For ideal rolling without slipping, \(v = r\omega\) needs to hold; this relationship ensures the sphere's rotation matches its translational motion.
Thus, understanding and calculating angular velocity is crucial here, as it determines the point at which the sphere shifts from sliding to rolling smoothly.
Rolling Without Slipping
Rolling without slipping occurs when the rolling object's point in contact with the surface does not slide across the surface.It describes a motion where the sphere rolls perfectly along the platform.For this to happen, there must be synchronization between translational and rotational motion.
  • The condition \(v = r \omega\) marks the transition to rolling without slipping, where the sphere's angular and linear velocities align.
  • At this stage, the sphere's center of mass moves with the same velocity as the platform, ensuring no relative motion or slippage.
  • Kinetic friction helps achieve this state by providing the necessary acceleration to equalize the rotational and transitional movements.
Achieving rolling without slipping simplifies the problem of object dynamics considerably since it means the energy is effectively transferred between translational and rotational forms without losses to slippage.

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Most popular questions from this chapter

A sphere is released on a smooth inclined plane from the top. When it moves down, its angular momentum is (1) conserved about every point (2) conserved about the point of contact only (3) conserved about the centre of the sphere only (4) conserved about any point on a line parallel to the inclined plane and passing through the centre of the ball

A uniform box of height \(2 \mathrm{~m}\) and having a square base of side \(1 \mathrm{~m}\), weight \(150 \mathrm{~kg}\), is kept on one end on the floor of a truck. The maximum speed with which the truck can round a curve of radius \(20 \mathrm{~m}\) without causing the block to tip over is (assume that friction is sufficient so that there is no sliding). (1) \(15 \mathrm{~m} / \mathrm{s}\) (2) \(10 \mathrm{~m} / \mathrm{s}\) (3) \(8 \mathrm{~m} / \mathrm{s}\) (4) depends on the value of coefficient of friction

Inner and outer radii of a spool are \(r\) and \(R\), respectively. A thread is wound over its inner surface and spool is placed over a rough horizontal surface. Thread is pulled by a force \(F\) as shown in figure. In case of pure rolling, which of the following statements are false? (1) Thread unwinds, spool rotates anticlockwise and friction acts leftwards. (2) Thread winds, spool rotates clockwise and friction acts leftwards. (3) Thread winds, spool moves to the right and friction acts rightwards. (4) Thread winds, spool moves to the right and friction does not come into existence.

Two bodies with moments of inertia \(I_{1}\) and \(I_{2}\left(I_{1}>I_{2}\right)\) have equal angular momenta. If their kinetic energies of rotation \(\operatorname{are} E_{1}\) and \(E_{2}\), respectively, then (i) \(E_{1}=E_{2}\) (2) \(E_{1}E_{2}\) (4) \(E_{1} \geq E_{2}\)

A yo-yo is placed on a rough horizontal surface and a constant force \(F\), which is less than its weight, pulls it vertically. Due to this (1) friction force acts towards left, so it will move towards left (2) friction force acts towards right, so it will move towards right (3) it will move towards left, so friction force acts towards left (4) it will move towards right so friction force acts towards right
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